Good efficiency with little hardware complexity is desirable for the operation of an electronically commutated motor.
The efficiency of a motor is defined byEfficiency=Pout/Pin  (1)
When the efficiency is at its maximum, the quotient Pout/Pin must therefore also be at its maximum.
In the above equation,Pin=U*I=electrical power absorbed by the motor  (2)Pout=T*n=mechanical shaft power output  (3),
where    U=voltage    I=current    T=torque    n=rotation speed.
At a constant load torque T=constant and constant rotation speed n=constant, i.e. in a state of constant load that exists, for example, in the case of a fan in continuous operation, the variable component in equation (1) is the absorbed power level Pin=U*I.
The voltage Û is normally constant, and the current I is thus the variable to be controlled.
The definition of the so-called air-gap torque isTMi(t)=CM*Ψ(t)*I(t)  (4)where    TMi=internal torque or air-gap torque of the motor    CM=machine constant    ψ=concatenated flux    I=current to the stator, e.g. current in one phase, or total current to the stator, as explained below.
The requirement that the curve for TMi be uniform, or “smooth,” yields the requirement that both the concatenated flux ψ and the current I should be sinusoidal. This results in the requirement that the phase relationship between current I and flux ψ be determined so that a maximum torque TMi is obtained.
If the stator flux ψ and stator current I in a three-phase synchronous motor are parallel vectors, the torque T generated by the motor is then equal to zero. If, on the other hand, the space vector is at right angles to the stator current, a maximum torque is then produced. This is similar to the situation with a direct-current motor.
Generating this right angle by a control procedure requires a control loop with feedback to the machine, indicating the position of the rotor. This feedback has often been implemented in synchronous machines using three Hall sensors. Today, in most cases, encoders (resolvers), optical incremental and absolute value sensors, or inductive sensors are used. Sensorless control systems can be carried out, in a context of block commutation, by measuring the back-EMF induced in the motor.
It is known, from the prior art, to operate a three-phase synchronous motor with good efficiency using field-oriented control (FOC). As depicted in FIG. 8, in this case the rotor position, and thus the phase relationship of the flux, is ascertained either via a rotor position sensor or using sensorless methods, e.g. a so-called “observer” design.
In field-oriented control (FOC), the measured phase currents are broken down by matrix operations (Park-Clarke transformation or inverse Park-Clarke transformation) into two components: a field-forming part id and a torque-forming part iq. This type of subdivision into components makes it possible in FOC to modify or control the field-forming variable id independently of the torque-forming variable iq. The field-forming variable is equal to zero at the point of maximum efficiency. This results in a special case that can easily be implemented with no need to carry out complex matrix operations, i.e. FOC can be dispensed with, in this special case.
Because matrix operations are not necessary, a simple microprocessor can be used, whereas expensive microprocessors having a digital signal processor (DSP) would otherwise be required for FOC.
In this case, a brief measurement operation can be used to determine the phase relationship between flux ψ and motor current I which results in the maximum torque T. The equation is:
      F    ⁡          (      x      )        =                    ∫        0        π            ⁢                        sin          ⁡                      (            x            )                          *                  sin          ⁡                      (                          x              +              α                        )                                =                                        cos            ⁡                          (              α              )                                *          π                2            =              max        .            
where:    x=rotation angle of rotor, usually measured in radians    α=phase difference between current I and flux ψ (see FIG. 1)
1. Necessary:
            f      ′        ⁡          (      α      )        =                    ⅆ                  ⅆ                      (            α            )                              ⁢                                    cos            ⁡                          (              α              )                                *          π                2              =                  0        ⇒                                            -                              sin                ⁡                                  (                  α                  )                                                      *            π                    2                    =                        0          ⇒          α                =        0            
2. Sufficient:
                              f          ′′                ⁡                  (          α          )                    =                                    ⅆ                          ⅆ                              (                α                )                                              ⁢                                                    -                                  sin                  ⁡                                      (                    α                    )                                                              *              π                        2                          >        0              ⁢                      for    ⁢                      α    =                  0        ⇒                              f            ′′                    ⁡                      (            0            )                              =                                                  cos              ⁡                              (                0                )                                      *            π                    2                =                  π          2                    
This calculation yields the requirement for a sinusoidal current, which must be in-phase with the concatenated flux ψ, in order for efficiency to become optimal.
This is illustrated in FIGS. 1 and 2. The number 20 illustrates the overlap between a phase current, e.g. i_U, and the variable ψ. It is evident that the area 20 reaches its maximum when ψ and i_U are in-phase.
If the flux ψ and current I deviate from the sinusoidal shape, FIG. 1 likewise indicates the need for the in-phase condition, in order to obtain improved efficiency.